Question
Classify the following functions as injection, surjection or bijection:
f : R → R, defined by f(x) = x3 - x
f : R → R, defined by f(x) = x3 - x
$\because\ \text{x}^2+\text{xy}+\text{y}^2\geq0$
$\Rightarrow\ \text{x}^2+\text{xy}+\text{y}^2-1\geq-1$
$\therefore\ \text{x}^2+\text{xy}+\text{y}^2-1\neq0$
$\Rightarrow \text{x}-\text{y}=0\Rightarrow \text{x}=\text{y}$ $\therefore$ f is one-one. Surjective: Let $\text{y}\in\text{R},$ then f(x) = y ⇒ x3 - x - y = 0 We know that a degree 3 equation has atleast one real solution. Let $\text{x}=\alpha$ be that real solution $\therefore\ \alpha^2-\alpha=\text{y}$ $\Rightarrow\ \text{f}(\alpha)=\text{y}$ $\therefore$ For each $\text{y}\in\text{R,}$ there exist $\text{x}=\alpha\in\text{R}$ such that $\text{f}(\alpha)=\text{y}$ $\therefore$ f is onto.Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.